5. Projections

Projection is the process of converting a three-dimensional object into a two-dimensional representation on a display screen. Conceptually, it answers one question: from where and how are we looking at a 3D world? Projection removes one spatial dimension (depth) while preserving meaningful visual information required for understanding shape, size, and position.

            flowchart TD
                Proj[Projections] --> Parallel[Parallel]
                Proj --> Pres[Perspective]
                Parallel --> Ortho[Orthographic]
                Parallel --> Oblique[Oblique]
                Pres --> P1[1-Point]
                Pres --> P2[2-Point]
                Pres --> P3[3-Point]
                Ortho --> Top[Top/Plan]
                Ortho --> Front[Front/Elevation]
                Oblique --> Cav[Cavalier]
                Oblique --> Cab[Cabinet]
            

Mathematically, projection maps a 3D point $(x, y, z)$ to a 2D point $(x', y')$ using geometric rules defined by the viewing method.

5.1 Parallel Projection

In parallel projection, all projection lines (projectors) are parallel to each other and perpendicular or inclined to the projection plane. There is no concept of a viewpoint or eye position.

Real-world analogy: looking at an object from infinitely far away, so perspective effects disappear.

• No depth perception: distant and near objects appear the same size.
• Parallel lines remain parallel: no convergence.
• Used for accuracy: preferred in engineering drawings and CAD.

5.1.1 Orthographic Projection

Orthographic projection is a special case of parallel projection where projection lines are perpendicular to the projection plane.

Only two coordinates are retained; depth is discarded.

Projection equations:

$$x' = x$$

$$y' = y$$

• True dimensions preserved: measurements remain exact.
• Multiple views: front, top, side views used together to understand full shape.
• Engineering usage: blueprints, mechanical design.

5.1.2 Oblique Projection

Oblique projection is a parallel projection where projection lines strike the projection plane at an angle instead of perpendicularly.

It allows one face of the object to appear in true shape while the depth is shown at an angle.

• Front face undistorted
• Depth visible, but not realistic

5.1.2.1 Cavalier Projection

In cavalier projection, the depth axis is projected at full scale, typically at $45^\circ$.

Depth scaling factor = 1

$$x' = x + z \cos \theta$$

$$y' = y + z \sin \theta$$

• Exaggerated depth: looks elongated.
• Simple to construct, but visually unrealistic.

5.1.2.2 Cabinet Projection

Cabinet projection improves realism by reducing depth by half.

Depth scaling factor = $\frac{1}{2}$

$$x' = x + \frac{z}{2}\cos \theta$$

$$y' = y + \frac{z}{2}\sin \theta$$

• Better visual balance
• Used in furniture and interior design sketches

5.2 Perspective Projection

Perspective projection models human vision. All projectors converge at a single point called the Center of Projection (COP).

Real-world analogy: how a camera or human eye sees the world.

• Depth perception present
• Parallel lines converge at vanishing points
• Distant objects appear smaller

5.2.1 Core Concepts

• COP (Center of Projection): eye or camera position
• Projection Plane: screen where image is formed
• Vanishing Point: point where parallel lines meet

5.2.2 Types of Perspective Projection

• 1-Point Perspective: one vanishing point; one set of parallel lines converges
• 2-Point Perspective: two vanishing points; used for corner views
• 3-Point Perspective: three vanishing points; includes height (skyscraper views)

5.2.3 Basic Perspective Projection Formula

Derived using similar triangles.

If COP is at distance $d$ from the projection plane:

$$x' = \frac{x \cdot d}{z}$$

$$y' = \frac{y \cdot d}{z}$$

• $z$ increases → object shrinks
• Non-linear scaling: realism emerges naturally

Projection Perspectives

• Projection: Mapping 3D coordinates \((x,y,z)\) to 2D plane \((x',y')\); loss of depth dimension (\(z\)).
• Parallel Projection: Projectors are parallel; preserves relative proportions; used in engineering/CAD.
  • Orthographic: Projectors perpendicular to plane; true dimensions preserved (\(x'=x, y'=y\)).
  • Oblique: Projectors inclined; preserves front face but angles depth.
• Cavalier vs. Cabinet: Cavalier uses full depth scale (distorted); Cabinet uses half depth scale (more realistic).
• Perspective Projection: Projectors converge at Center of Projection (COP); simulates human vision.
• Vanishing Points: Where parallel lines appear to meet in perspective (1, 2, or 3 points).
• Perspective Formula: \(x' = x \cdot (d/z)\); objects shrink as distance (\(z\)) increases (foreshortening).